Coupled cluster computations with two-body currents Gaute Hagen - - PowerPoint PPT Presentation

Coupled cluster computations with two-body currents

Gaute Hagen Oak Ridge National Laboratory

SFB1044 workshop: Electromagnetic observables for low-energy nuclear physics Mainz, October 2nd, 2018

70 60 80

Ab-initio Method: Solve A- nucleon problem with controlled approximations and systematically improvable. Realistic: BEs within 5% of experiment and starts from NN + 3NFs

Explosion of many-body methods (Coupled clusters, Green’s function Monte

Carlo, In-Medium SRG, Lattice EFT, MCSM, No-Core Shell Model, Self-Consistent Green’s Function, UMOA, …)

Application of ideas from EFT and renormalization group (Vlow-k, Similarity

Renormalization Group, …)

Trend in realistic ab-initio calculations

100Sn

Coupled-cluster method (CCSD approximation)

Ansatz: Correlations are exponentiated 1p-1h and 2p-2h excitations. Part of np-nh excitations included! Coupled cluster equations J Scales gently (polynomial) with increasing problem size o2u4 . J Truncation is the only approximation. J Size extensive (error scales with A) L Most efficient for closed (sub-)shell nuclei

Alternative view: CCSD generates similarity transformed Hamiltonian with no 1p-1h and no 2p-2h excitations.

Oxgyen chain with interactions from chiral EFT

Hebeler, Holt, Menendez, Schwenk, Annu. Rev. Nucl. Part. Sci. 65, 457 (2015)

N3LO(EM) + 3NF(Local, Λ3N =400MeV) Measured at RIKEN

Continuum!

Challenge: Collectivity and transition strengths

§ 14C computed in FCI and CC with psd effective interaction § Neutron effective charge of charge = 1 § Need excitations beyond 4p4h to describe B(E2) even if 2+ energy is reproduced

Nph

14C

Challenge: Collectivity and transition strengths

§ 14C computed in FCI and CC with psd effective interaction § Neutron effective charge of charge = 1 § Need excitations beyond 4p4h to describe B(E2) even if 2+ energy is reproduced

§ Developing higher orders and higher rank (3NF, 4NF) [Epelbaum 2006; Bernard et al 2007; Krebs et al 2012; Hebeler et al 2015; Entem et al 2017, Reinert et al 2017…] § Propagation of uncertainties on the horizon [Navarro Perez 2014, Carlsson et al 2015] § Different optimization protocols [Ekström et al 2013, Carlsson et al 2016] § Improved understanding/handling via SRG [Bogner et al 2003; Bogner et al 2007] § local / semi-local / non-local formulations [Epelbaum et al 2015, Gezerlis et al 2013/2014] § Chiral EFT’s with explicit Delta isobars [Krebs et al 2018, Piarulli et al 2017, Ekstrom et al 2017]

Nuclear forces from chiral effective field theory

[Weinberg; van Kolck; Epelbaum et al.; Entem & Machleidt; …]

Nuclear forces from chiral effective field theory

[Weinberg; van Kolck; Epelbaum et al.; Entem & Machleidt; …]

− 1 4

From Sofia Quaglioni and Kyle Wendt

1.8/2.0(EM): Accurate BEs Soft interaction: SRG NN from Entem & Machleidt with 3NF from chiral EFT

• K. Hebeler et al PRC (2011).
• T. Morris et al, arXiv:1709.02786

(2017).

NNLOsat: Accurate radii and BEs § Simultaneous optimization of NN and 3NFs § Include charge radii and binding energies of 3H, 3,4He, 14C, 16O in the optimization § Harder interaction: difficult to converge beyond 56Ni

• A. Ekström et al, Phys. Rev. C 91, 051301(R) (2015).

A family of interactions from chiral EFT

Neutron radius and skin of 48Ca

Uncertainty estimates from family of chiral interactions:

• K. Hebeler et al PRC (2011)

DFT: SkM*, SkP, Sly4, SV-min, UNEDF0, and UNEDF1

• Neutron skin significantly

smaller than in DFT

• Neutron skin almost

independent of the employed Hamiltonian

• Our predictions for 48Ca are

consistent with existing data

• G. Hagen et al, Nature Physics

12, 186–190 (2016) 0.15 0.18 0.21

Rskin (fmD

3.2 3.3 3.4 3.5

Rp (fmD A

3.4 3.5 3.6

Rn (fmD B

2.0 2.4 2.8

αD (fmn D C

1.8/2.0 (EM)

0.15 0.18 0.21

Rskin (fmD

3.2 3.3 3.4 3.5

Rp (fmD A

3.4 3.5 3.6

Rn (fmD B

2.0 2.4 2.8

αD (fmn D C

• DFT results are consistent and

within band of ab-initio results

• αD meausred by the Osaka-

Darmstadt collaboration

• Ab-initio prediction overlaps

with experimental uncertainty

• αD constrains the neutron

skin to 0.14 – 0.20fm

Ab-initio prediction from correlation with Rp: 2.19 ≲ αD ≲ 2.60 fm3

Dipole polarizability of 48Ca

• J. Birkhan et al PRL (2017)

0.15 0.18 0.21

Rskin (fmD

3.2 3.3 3.4 3.5

Rp (fmD A

3.4 3.5 3.6

Rn (fmD B

2.0 2.4 2.8

αD (fmn D C

0.15 0.18 0.21

Rskin (fmD

3.2 3.3 3.4 3.5

Rp (fmD A

3.4 3.5 3.6

Rn (fmD B

2.0 2.4 2.8

αD (fmn D C

• G. Hagen et al, Nature Physics

12, 186–190 (2016)

• J. Birkhan et al PRL (2017)

Compute the dipole polarizability of

48Ca with increased precision

• Triples impacts 𝛽%
• Less than 1% effect

from triples on radii

• The inclusion of triples

fragments the strength and increases strength at higher energies

• Triples impacts the

running sum for 𝛽% Higher order corrleations are important!

• M. Miorelli et al,
• Phys. Rev. C 98, 014324 (2018)

15% 6%

Compute the dipole polarizability of

48Ca with increased precision

15% 6%

• Triples impacts 𝛽%
• Less than 1% effect

from triples on radii

• The inclusion of triples

fragments the strength and increases strength at higher energies

• Triples impacts the

running sum for 𝛽% Higher order corrleations are important!

• M. Miorelli et al,
• Phys. Rev. C 98, 014324 (2018)

Coulomb Sum Rule

The CSR is the total integerated strength of inelastic longitudinal response function Here 𝜍 𝑟 is the nuclear charge operator Final state different from g.s. since we want the inelastic response We approached the problem as we do for the calculation of the total strength of the dipole response function in PRL 111, 122502 (2013).

Inclusive electron scattering and the Coulomb sum rule

Towards 𝜉-scattering/response of 16O and 40Ar: Electron scattering off 16O and 40Ca

15

15

4He 4He

Benchmark with “exact” Hyperspherical Harmonics for 4He

Coulomb Sum Rule

Inclusive electron scattering and the Coulomb sum rule

16

§ Good agreement in 4He § CSR for 16O based on NNLOsat and N3LO(EM) § Comparison to data in 12C and to Mihaila and Heisenberg (PRL 2000)

Comparison to data in 4He and 16O

Comparison to data in 40Ca with NNLOsat

§ Excellent agreement with elastic charge form factor up to momentum transfers of ~500MeV/c § Very little data for the CSR § To exhaust the sum rule need to integrate longitudinal response

• ver large energy range

Data from Ingo Sick

A 50 year old problem: The puzzle of quenched of beta decays

Quenching obtained from charge- exchange (p,n) experiments. (Gaarde 1983).

§ Renormalizations of the Gamow-Teller operator? § Missing correlations in nuclear wave functions? § Model-space truncations? § Two-body currents (2BCs)?

• G. Martinez-Pinedo et al, PRC 53, R2602 (1996)

Theory to experiment ratios for beta decays in light nuclei from NCSM

N4LO(EM ) + 3Nlnl SRG-evolved to 2.0fm-1 (cD = -1.8)

Entem, Machleidt & Nosyk, PRC 96, 024004 (2017) In QMC calculations of beta-decays 2BC increase the GT strength by 2-3%

• S. Pastore et al, PRC 97, 022501 (2018).

0.95 1.00 1.05 1.10 ratio to experiment

14O0 →14 N1 10C0 →10 B1 7Be 3

2 →7 Li 3 2

7Be 3

2 →7 Li 1 2

6He0 →6 Li1 3H 1

2 →3 He 1 2

GT only GT + 2BC

Theory to experiment ratios for beta decays in light nuclei from NCSM

0.95 1.00 1.05 1.10 ratio to experiment

14O0 →14 N1 10C0 →10 B1 7Be 3

2 →7 Li 3 2

7Be 3

2 →7 Li 1 2

6He0 →6 Li1 3H 1

2 →3 He 1 2

GT only GT + 2BC

NNLOsat (cD = 0.82)

100Sn – a nucleus of superlatives

§ Heaviest self-conjugate doubly magic nucleus § Largest known strength in allowed nuclear β-decay § Ideal nucleus for high-

• rder CC approaches

Hinke et al, Nature (2012)

Quantify the effect of quenching from correlations and 2BCs

Faestermann, Gorska, & Grawe(2013)

t=4

Coupled cluster calculations of beta-decay partners

𝑆+ = - 𝑠/

0 𝑞0 2𝑜/ + 1

4 - 𝑠/5

06 𝑞0 2𝑂 6 2𝑂 5𝑜/ + 1

36 - 𝑠/5:

06; 𝑞0 2𝑂 6 2𝑂 ; 2𝑂 :𝑂 5𝑜/

H = e−T HNeT

Diagonalize via a novel equation-of-motion technique:

• A. Ekström, G. Jansen, K. Wendt et al, PRL

113 262504 (2014)

H = e−T HNeT

Diagonalize via a novel equation-of-motion technique:

Coupled cluster calculations of beta-decay partners

𝑆+ = - 𝑠/

0 𝑞0 2𝑜/ + 1

4 - 𝑠/5

06 𝑞0 2𝑂 6 2𝑂 5𝑜/ + 1

36 - 𝑠/5:

06; 𝑞0 2𝑂 6 2𝑂 ; 2𝑂 :𝑂 5𝑜/

• A. Ekström, G. Jansen, K. Wendt et al, PRL

113 262504 (2014)

Charge exchange EOM-CCSDT-1

§ Bloch-Horowitz is exact; iterative solution poss. § Q-space is restricted to: § No large memory required for lanczos vectors § Can only solve for one state at a time § Reduces matrix dimension from ~109 to ~106

• W. C. Haxton and C.-L. Song Phys. Rev. Lett. 84 (2000); W. C. Haxton Phys. Rev. C 77, 034005 (2008)
• C. E. Smith, J. Chem. Phys. 122, 054110 (2005)

P-space Q-space

𝐼 =>>?%@AB = 𝑇 𝐼 = 𝑇 𝐸 𝐼 = 𝑇 𝑈 𝑊 𝑇 𝑇 𝐼 = 𝐸 𝐸 𝐼 = 𝐸 𝑈 𝑊 𝐸 𝑇 𝑊 𝑈 𝐸 𝑊 𝑈 𝑈 𝐺 𝑈

˜ Epqr = ˜ ep + ˜ eq + ˜ er ≤ ˜ E3max

100In from charge exchange coupled-cluster

equation-of-motion method

Charge-exchange EOM-CC with perturbative corrections accounting for excluded 3p3h states:

Hinke et al, Nature (2012)

1.8/2.0 (EM)

Normal ordered one- and two-body current

Gamow-Teller matrix element:

Normal ordered operator:

ˆ OGT = O + O1

N + O2 N

Benchmark between NCSM and CC using NN-N4LO 3Nlnl in 8He:

Normal ordered one- and two-body current

Gamow-Teller matrix element:

Normal ordered operator:

ˆ OGT = O + O1

N + O2 N

Benchmark between NCSM and CC using NN-N4LO 3Nlnl and NNLOsat :

14O0 →14 N1

|MGT|2

Super allowed Gamow-Teller decay of 100Sn

GT: 7-11 MEC+GT: 5-7

Role of 2BC and correlations in 100Sn

The small role of short-ranged 2BC on GT decay

PRL 107, 062501 (2011)

• J. Menéndez, D. Gazit, A. Schwenk

One-body normal ordering of 2BC in free Fermi gas

Short-ranged contact term of 2BC (heavy meson exchange)

@ ORNL / UTK: G. R. Jansen, T. Morris, T. Papenbrock, Z. H. Sun @ MSU: W. Nazarewicz @ Chalmers: A. Ekström, C. Forssén @ Hebrew U: N. Barnea @ INT: S. R. Stroberg @ Mainz: S. Bacca, C. Payne, J. Simonis @ MSU/ U Oslo: M. Hjorth-Jensen @ Trento: G. Orlandini @ TRIUMF: P. Gysbers, J. Holt, P. Navratil @ TU Darmstadt: C. Drischler, C. Stumpf, K. Hebeler, R. Roth, A. Schwenk @ LLNL: K. Wendt, S. Quaglioni

Collaborators